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(a) 2(90 ) 5 (b) (c) (d)
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36 ) 120 151 2 15 30r
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1 10 (70 1 10 (20r)
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20r)
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CHAPTER 1 Lines, Angles, and Triangles
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1.8 Finding rotations In a half hour, what turn or rotation is made (a) by the minute hand, and (b) by the hour hand of a clock What rotation is needed to turn (c) from north to southeast in a clockwise direction, and (d) from northwest to southwest in a counterclockwise direction (see Fig. 1-24)
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Fig. 1-24
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Solutions
(a) In 1 hour, a minute hand completes a full circle of 360 . Hence, in a half hour it turns 180 . (b) In 1 hour, an hour hand turns
1 12
of 360 or 30 . Hence, in a half hour it turns 15 . 45 135 .
1 4 (360
(c) Add a turn of 90 from north to east and a turn of 45 from east to southeast to get 90 (d) The turn from northwest to southwest is ) 90 .
1.9 Finding angles Find the measure of the angle formed by the hands of the clock in Fig. 1-25, (a) at 8 o clock; (b) at 4:30.
Fig. 1-25
Solutions
(a) At 8 o clock, m/a (b) At 4:30, m/ b
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1.10 Applying angle facts In Fig. 1-26, (a) name two pairs of perpendicular segments; (b) find m/a if m/b m/AEB and m/CED.
42 ; (c) find
Fig. 1-26
CHAPTER 1 Lines, Angles, and Triangles
Solutions
(a) Since /ABC is a right angle, AB ' BC. Since /BEC is a right angle, BE ' AC. (b) m/a (c) m/AEB 90 m/b 180 90 42 180 48 . 90 90 . m/CED 180 m/1 180 45 135 .
m/BEC
1.6 Triangles
A polygon is a closed plane figure bounded by straight line segments as sides. Thus, Fig. 1-27 is a polygon of five sides, called a pentagon; it is named pentagon ABCDE, using its letters in order.
Fig. 1-27
A quadrilateral is a polygon having four sides. A triangle is a polygon having three sides. A vertex of a triangle is a point at which two of the sides meet. s (Vertices is the plural of vertex.) The symbol for triangle is n ; for triangles, n. A triangle may be named with its three letters in any order or with a Roman numeral placed inside of it. Thus, the triangle shown in Fig. 1-28 is n ABC or n I; its sides are AB, AC, and BC; its vertices are A, B, and C; its angles are /A, /B, and /C.
Fig. 1-28
1.6A Classifying Triangles
Triangles are classified according to the equality of the lengths of their sides or according to the kind of angles they have. Triangles According to the Equality of the Lengths of their Sides (Fig. 1-29) 1. Scalene triangle: A scalene triangle is a triangle having no congruent sides.
Thus in scalene triangle ABC, a b c. The small letter used for the length of each side agrees with the capital letter of the angle opposite it. Also, means is not equal to.
2. Isosceles triangle: An isosceles triangle is a triangle having at least two congruent sides.
Fig. 1-29
CHAPTER 1 Lines, Angles, and Triangles
Thus in isosceles triangle ABC, a c. These equal sides are called the legs of the isosceles triangle; the remaining side is the base, b. The angles on either side of the base are the base angles; the angle opposite the base is the vertex angle.
3. Equilateral triangle: An equilateral triangle is a triangle having three congruent sides.
Thus in equilateral triangle ABC, a b c. Note that an equilateral triangle is also an isosceles triangle.
Triangles According to the Kind of Angles (Fig. 1-30) 1. Right triangle: A right triangle is a triangle having a right angle.
Thus in right triangle ABC, /C is the right angle. Side c opposite the right angle is the hypotenuse. The perpendicular sides, a and b, are the legs or arms of the right triangle.
2. Obtuse triangle: An obtuse triangle is a triangle having an obtuse angle.
Thus in obtuse triangle DEF, /D is the obtuse angle.
3. Acute triangle: An acute triangle is a triangle having three acute angles.
Thus in acute triangle HJK, /H, /J, and /K are acute angles.
Fig. 1-30
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